Music Trade Review

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THE
MUSIC TRADE
REVIEW
13
shows their substantial correctness, but it seems
to me that there is little necessity to carry out
calculations for tension to such exactness as is
indicated by quotients running to twenty-two
places. Regarding the statements as to errors in
Conducted and Edited by Wm. B. White.
scaling and the resulting bad "breaks" between
I am very glad ot the opportunity to present to the pitch of C.,, the wire being No. 16, and the the bass and treble, it is a fact that many pianos
readers of this department a very interesting length 14 inches. By my calculations the ten- will be found to have their highest pitched bass
letter from E. M. Payson, who for many years sion of a 1-inch string of the given pitch and strings longer than the plain strings immediately
has been the head tuner for the John Church thickness is .7787643+, etc. Therefore, the tension above them. According to previous calculations
Co.'s Chicago branch. This letter may be one for a 14-inch string will be obtained by multi- of my own, the bass strings nearest to the plain
of those exceptions that are so erroneously sup- plying this result by the square of 14, that is, wire should be shorter by about 10 per cent.
posed to "prove the rule;" at any rate we have 196; whence we have the required tension on'
Doubtless it will be easily understood by all
here a contribution of genuine value to the stu- the string, which is 152.6378116-)-, etc., this being readers that the tension of 200 lbs. is as much
dent of the theory of piano construction, and the number of pounds of tension required for the too great for the highest C string as that of 87
this sort of thing is very seldom investigated by string under the given conditions.
lbs. is too little. As a general thing the tension
a practical tuner. For a long time 1 have been
"For practical purposes it is unnecessary to •for the highest strings should be about 135 lbs.,
preaching to readers of the department the de- carry out the calculation to twenty-two places as this being calculated for strings strung with
sirability of a closer investigation on their part I have done throughout, but in my calculations I 13 wire and at 2 inches of length for the highest
of the theory of construction and also of the have found it better to allow no possibility of
C, and so on proportionately. And in this re-
acoustics of music. Many practical tuners have error. I have also worked out the tension for any gard it might be well for the reader to note that
at one time or another taken upon themselves interval of the octave of tempered sounds, and in calculating string lengths there is one very
to demonstrate (to their complete satisfaction)
my accumulated error is less than one two good and simple rule that will always give the
that devotion to this sort of study is neither billion-billionth of a pound, so you can see the octave lengths with absolutely accurate propor-
necessary nor useful for the pianomaker or tuner, exhaustive accuracy of the manner in which I tion, and another from which all intermediate
and that such a man does not need to have any have handled the subject. I have also carried
lengths may be calculated. To obtain the true
knowledge of the scientific aspects of piano calculations of vibration frequency to the same length for a string one octave lower than a given
building. That this idea is totally at variance number of decimal places.
string, multiply the length of the given string
with my own notions on the subject it is hardly
"I have likewise worked out a table for 420 by 1.9375. This will give the octave lengths from
necessary for me to say here.' So many people, combinations of iron and copper wrapping wire the highest C down to the end of the plain wire
however, have held that such is the case, and whereby I can tell just the required weight of a strings. Now to obtain the intermediate string
that the "practical men" are right, that it is a 1-inch covered string. Knowing that, I can there- lengths, proceed as follows. Given the length of
genuine relief to note that one of the "practical fore calculate the proper tension for all bass one string, to find the length of the string which
men" at least agrees with me.
strings. . . . Such tables as these have never, I be- shall produce any number of semi-tones above
Mr. Payson's very illuminating communication lieve, been worked out before. . . . It is most inter- or below the pitch of the given string, all other
states that he has been a student of the piano esting to compare with this unit of measurement pertinent conditions remaining equal:
for twenty-five years, and that he has tuned over tt>e different variations shown by instruments.
1. Take the logarithm of the length of the
thirty-five thousand pianos in that time. With- The monstrosities of some scales are certainly given string.
out quoting; the whole of his letter, we reproduce marvelous. . . . I have noticed one piano wherein
2. Multiply the number .025086 by the number
here the salient portions thereof. Mr. Payson, the highest C string had a length of 2.25 inches of semi-tones that the sound to be given by the
then, says in part "The art of scientific scale and was of number 14 wire, which therefore re- required string is above or below the sound of
drawing, more especially as concerns string quired a tension of 200.8 lbs. to give a frequency the given string.
of 4,138.44 vibrations per second. The other for
r lengths, thickness and tensions, has been a hobby
3. If the required string is below the given
of mine for some years, and I have worked out the same pitch was 1.75 inches long and the wire string, add together the two numbers obtained;
a most exhaustive table, whereby, given the was number 11}4. requiring a tension of only if it be above, subtract the second number from
length, size and desired pitch for any string I 87.32 lbs. It is needless to say that in the latter- the first; the result in either case is the loga-
can at once calculate the required tension.
case the tone was badly wanting, and in the rithm of the required length.
"The basis of calculation was derived from former the limit of tensile strength was nearly
For example, suppose the length of middle C
nearly one hundred tests on wire of number 16% reached and the factor of safety consequently string to be 2S.82 inches; it is required to find
size (Poehlman gauge), at a temperature of 68 very small. On this piano the C above middle C the proper length of the string for C sharp, one
degrees Fahrenheit, and by taking the average was nearly 15 inches long, at least an inch longer semi-tone above.
of results gained. This was done on a fourteen- tkan any I have ever seen. It is impossible to
Length of C string in hundnvlfhs of an inch -= 2882.
45943 r= IOK. of 2SS2
get a piano with even tone quality when the ten-
inch string of most accurate measurement.
02508 ^ iho number .02508 (G) multiplied by the num-
ber (1) of semitones that required string
"From this I have been able to calculate the sion is disregarded, and this is the reason why
is above given strinc.
proper tension of all the 88 tones on a piano, for so many pianos have such bad 'breaks' between
By subtraction 4'A4Xr, - Lop. of 2718 --=-= length of re-
quired string in hunrirrdths of an inch «
the
plain
and
wound
strings.
.
.
.
The
best'break'I
all the sizes of wire used, from No. 12 to >*o. 26,
27.18 inches.
including the intermediate half sizes, and at the have ever seen is to be found on an old grand
By reversing this process and adding instead
piano of a famous make, which has been used
international tempered scale pitch (A 435).
of subtracting, we obtain the proper lengths for
"This required 1404 different calculations and very hard for fifteen years, so that the voicing strings bnlow, and thus all intermediate strings
each was carried out to twenty-two places of has entirely disappeared, as it were. Yet the above or below octave strings may be calculated
decimals, and then proved. These calculations evenness at the 'break' is remarkable. The 'sci- for with accuracy. This rule was first suggested
are on one inch of length, and as the pitch of a entifically' drawn scales are truly rare. . . . by the late Prof. Pole, F. R. S., of London. The
"If you ever read this letter as far as this
string varies as the square of its length, the
value of any logarithm or the logarithm of any
required tensions for any given string can be point—please forgive its length—but the subject number to the base 10 or to the base "e" can be
is
of
such
interest
to
me
that
I
feel
as
did
the
found by multiplying the tension for a one-inch
obtained from any set of logarithmic tables. I
string of the given thickness and pitch as given lunatic who, when asked if he knew the differ- would certainly suggest to all who are interested
ence
between
a
horse
and
a
hobby,
replied:
'Well,
in the calculations, by the square of the actual
in such matters the propriety of a preliminary
length of the given string. For example: It is you can get off of a horse.' . . . I believe that if any study of and familiarity with the use of loga-
other
living
man
has
carried
further
investiga-
required to find the tension for a string to give
rithms. These useful formulae greatly simplify
tions along this line, he has not yet given his
all arithmetical and algebraic calculations.
knowledge to the world.
* * * *
"Very truly yours,
Altogether, I am very glad of the opportunity
E. M. PAYSON."
afforded this week of delivering this little com-
mentary on the very able and painstaking work
I have found it necessary to condense Mr. of Mr. Payson. Readers should give great attenr
Payson's letter somewhat, as space was lacking, tion to this gentleman's letter, as it contains some
D O N ' T purchase any
but the salient points are all given above. It very valuable and well digested information.
new tools until you have
will be noted that Mr. Payson's decimal calcula- The neat little rule which I have noted for calcu-
tions are given here only to seven places, instead lating string lengths when the length of any one
consulted our catalogue!
of to twenty-two as in the original. The reason string is known, will be found to be perfectly
We make a specialty of
for this compression is that the reader certainly accurate, and is recommended to the attention
TUNERS' TOOLS, OUTFITS
will not desire to verify a calculation to twenty- of scale designers and students of the theory of
two places of decimals, and indeed the process construction.
and SUPPLIES at very
would be very tedious unless logarithms were
* • • •
reasonable prices.
used. I must say, however, that this most lucid
Communications for the department should be
communication stands as a model of acoustical
investigation. The man who is capable of this addressed to the Editor Technical Department
sort of work is a valuable man to the trade, and The Music Trade Review.
THE TUNERS' SUPPLY CO.
should be heard of further.
FRANKLIN SQUARE,
BOSTON, MASS.
Prof. Sparks has rented warerooms in Logan,
Regarding Mr. Payson's ' specific statements,
such verification as I have been able to make O., where he will handle a full line of pianos.
TUNERS!